Tristan Cook comments on Replicating and extending the grabby aliens model

[Tristan Cook]

Thanks for your response Robin.

I stand by the claim that both (updating on the time remaining) and (considering our typicality among all civilizations) is an error in anthropic reasoning, but agree there are non-time remaining reasons reasons to expect $n>3$ (e.g. by looking at steps on the evolution to intelligent life and reasoning about their difficulties). I think my ignorance based prior on $n$ was naive for not considering this.

I will address the issue of the compatibility of high $n$ and high $L_{max}$ by looking at the likelihood ratios of pairs of ($n$, $L_{max}$).

I first show a toy model to demonstrate that the deadline effect is weak (but present) and then reproduce the likelihood ratios from my Bayesian model.

My toy model (code here)

I make the following simplifications in this toy likelihood ratio calculation

• There are two types of habitable planets: those that are habitable for $5Gy$ and those that are habitable for $L_{max}>5Gy$

• Given a maximum habitable planet duration $L_{max}$, I suppose there are approximately $T\left(L_{max}\right)\approx 5\cdot {10}^{18}\cdot (L_{max}inGy{)}^{0.5}$ planets in the observable universe of this habitability.

• The universe first becomes habitable at 5 Gy after the Big Bang.

• All ICs become GCs.

• There are no delay steps.

• There are no try-once steps.^[1]

And the most important assumptions:

• If there are at least 5000 GCs^[2] $20Gy$ after the Big Bang I suppose a toy ‘deadline effect’ is triggered and no ICs or GCs appear after $20Gy$

• If there are at least 1000 GCs $10Gy$ after the Big Bang I suppose Earth-like life is precluded (these cases have zero likelihood ratio).

Writing $f\left(t\right)$ & $F\left(t\right)$for the PDF & CDF of the Gamma distribution parameters n and h:

• If $T\left(L_{max}\right)F\left(5\right)>1000$ then there are too many GCs and humanity is precluded.

  • The likelihood ratio is 0.

• If the first case is false and $T\left(L_{max}\right)F\left(15\right)+T\left(5\right)F\left(5\right)>5000$ the toy deadline effect is triggered and I suppose no more GCs arrive.

  • The likelihood ratio is directly proportional to $f\left(4\right)/\left(T\right(L_{max}\left)F\right(15)+T(5\left)F\right(5\left)\right)$

• Otherwise, there is no deadline effect and life is able to appear late into the universe.

  • The likelihood ratio is directly proportional, with the same constant of proportionality as above, to $f\left(4\right)/\left(T\right(L_{max}\left)F\right(L_{max})+T(5\left)F\right(5\left)\right)$

Toy model results

Above: some plot of the likelihood ratios of pairs of $(n,L_{max})$. The three plots vary by value of $h$ (the geometric mean of the hardness of the steps).

The white areas show where life on Earth is precluded by early arriving GCs.

The contour lines show the (expected) number of GCs at 20 Gy. For more than 1000 GCs at 20 Gy (but not so many to preclude human-like civilizations) there are relatively high likelihood ratios.

For example, $(h=1000Gy,n=7,L_{max}=1000Gy)$ has a likelihood ratio 1e-24. In this case there are around 5200 GCs existing by 20 Gy, so the toy deadline effect is triggered. However, this likelihood ratio is much smaller than the case $(h=1000Gy,n=7,L_{max}=10Gy)$ [just shifting along to the left] which has ratio 1e-20. In this latter case, there are 0.4 expected GCs by 20 Gy, so the toy deadline effect is not triggered.

Toy model discussion

This toy deadline effect that can be induced by higher $L_{max}$ (holding other parameters constant) is not strong enough to compete with humanity becoming increasingly atypical when$L_{max}$. Increasing $L_{max}$ makes Earthlike civilizations less typical by

(1) the count effect, where there are more longer lived planets than shorter lived ones like Earth. In the toy model, this effect accounts for $T\left(L_{max}\right)/T\left(5\right)\approx \sqrt{5/L_{max}}$ decrease in typicality. This effect is relatively weak.

(2) the power law effect, where the greater habitable duration allows for more attempts at completing the hard steps. When there is a deadline effect at 20 Gy (say) and the universe is habitable from 5 Gy, any planet that is habitable for at least 15 Gy has three times the duration for an IC or GC to appear. For sufficiently hard steps, this effect is roughly decreases life on Earth’s typicality by $(15/5{)}^n$. This effect is weaker when the deadline is set earlier (when there are faster moving GCs).

This second effect can be strong, dependent on when the deadline occurs. At a minimum, if the universe has been habitable since 5 Gy after the Big Bang the deadline effect could occur in the next 1 Gy and there would still be a $(10/5{)}^n$ typicality decrease. The power law effect’s decrease on our typicality can be minimised if all planets only became habitable at the same time as Earth.^[3]

This toy deadline effect model also shows that it only happens in a very small part of the sample space. This motivates (to me) why the posterior on $L_{max}$ is so pushed away from high values.

Likelihoods from the full model

I now show the likelihood ratios for pairs $(n,L_{max})$ having fixed the other parameters in my model. I set

• the probability of passing through all try-once steps $w=1$

• the parameter that controls the habitability of the early universe $u={10}^{-5}$

• the delay steps to take $1Gy$.

The deadline effect is visible in all cases. For example, the first graph in 1) with $h={10}^{4}Gy,v=c,f_{GC}=1$ has$(n=5,L_{max}={10}^{4}Gy)$ with likelihood ratio 1.5e-25. Although this is high relative to other likelihood ratios with $L_{max}={10}^{4}Gy$ it is small compared to moving to the left to $(n=5,L_{max}=5)$ which has likelihood ratio 3e-22.

1) $v=c,f_{GC}=1$

The plots differ by value of $h$

2) $v=c,f_{GC}=0.1$

3) $v=0.1c,f_{GC}=0.1$

4) $v=0.1c,f_{GC}=0.1$

1. ^

   For SSA-like updates, this does not in fact matter

2. ^

   Chosen somewhat arbitrarily. I don’t think the exact number matters though it should higher for slower expanding GCs.

3. ^

   This also depends on our reference class. I consider the reference class of observers in all ICs, but if we restrict this to observes in ICs that do not observe GCs then we also require high expansion speeds